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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ssbbi | Structured version Visualization version GIF version |
Description: Biconditional property for substitution, closed form. Specialization of biconditional. Uses only ax-1--5. Compare spsbbi 2390. (Contributed by BJ, 22-Dec-2020.) |
Ref | Expression |
---|---|
bj-ssbbi | ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biimp 204 | . . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) | |
2 | 1 | alimi 1730 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑥(𝜑 → 𝜓)) |
3 | bj-ssbim 31810 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓)) |
5 | biimpr 209 | . . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)) | |
6 | 5 | alimi 1730 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑥(𝜓 → 𝜑)) |
7 | bj-ssbim 31810 | . . 3 ⊢ (∀𝑥(𝜓 → 𝜑) → ([𝑡/𝑥]b𝜓 → [𝑡/𝑥]b𝜑)) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡/𝑥]b𝜓 → [𝑡/𝑥]b𝜑)) |
9 | 4, 8 | impbid 201 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 [wssb 31808 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 |
This theorem depends on definitions: df-bi 196 df-ssb 31809 |
This theorem is referenced by: bj-ssbbii 31813 |
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